Inference for GLM

Inference for GLM - Inference for the General Linear Model...

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Inference for the General Linear Model 1. Individual parameter tests. Assume we are testing the null hypothesis: 01 1 :0 a H H β = The t-test is generally used for hypothesis tests: 1 0 11 1 2 2 0 ˆ calc b i bb t s x σ −− == ⎛⎞ ⎜⎟ ⎝⎠ , where 1 b s is the standard error of the OLS estimate b 1 and we insert the specific value from the null hypothesis. We would only use the z-test if we had knowledge of the true value, 2 () Var u = . Of course, the t-distribution converges to the z-distribution, but we rarely have sufficient degrees of freedom to use the z-distribution. 2. Joint parameter test for the regression. Recall, from the simple linear model (SLM), that the hypothesis above could also be tested using an F -test. In the case of the SLM, the hypothesis above also becomes a hypothesis of whether the regression model explains a significant portion of the variation in the dependent variable. The question asked is: “Is the regression statistically significant?” The F is a t 2 , for the SLM the F is: 1 2 2 22 0 2 1 2 2 /(1) 0 ˆ (2 ) ˆ ii calc b i i bx t se n x = = ∑∑ The F statistic is a ratio of two variances, as shown in the final form above. If we expand
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Inference for GLM - Inference for the General Linear Model...

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